Designs and topology

We construct a face two‐colourable, blue and green say, embedding of the complete graph Kn in a nonorientable surface in which there are (n−1)/2 blue faces each of which have a hamilton cycle as their facial walk and n(n−1)/6 green faces each of which have a triangle as their facial walk; equivalently a biembedding of a Steiner triple system of order n with a hamilton cycle decomposition of Kn, for all n≡3(mod36) and n≠3. Using a variant of this construction, we establish the minimum genus of nonorientable embeddings of the graph K36k+3+Km¯, for m=18k+1+6s where k≥1 and 0≤s≤k−1.

K_{n}

.…”

Section: Introductionmentioning

confidence: 53%

K_{n}

? In this article we will show that, in the case where

n \equiv 3 (mod 36)

and

n > 3

, the answer is in the affirmative.…”

Section: Introductionmentioning

confidence: 99%

Biembedding a Steiner Triple System With a Hamilton Cycle Decomposition of a Complete Graph

McCourt

2013

“…There is a well-studied connection between graph embeddings and combinatorial designs arising from the fact that when a graph is embedded in a surface, the faces can be regarded as blocks of a design. An excellent survey on graph embeddings and combinatorial designs can be found in [14]. A face 2-colorable circular embedding of the complete graph K k (in some surface) in which each face is a triangle is a biembedding of two Steiner triple systems of order k. Results relating to biembeddings of Steiner triple systems can be found in [5], [19], [20].…”

Section: Introductionmentioning

confidence: 99%

Face 2‐Colorable Embeddings with Faces of Specified Lengths

Maenhaut

Smith

2015

Suppose M = m 1 , m 2 , . . . , m r and N = n 1 , n 2 , . . . , n t are arbitrary lists of positive integers. In this article, we determine necessary and sufficient conditions on M and N for the existence of a simple graph G, which admits a face 2-colorable planar embedding in which the faces of one color have boundary lengths m 1 , m 2 , . . . , m r and the faces of the other color have boundary lengths n 1 , n 2 , . . . , n t . Such a graph is said to have a planar (M; N )-biembedding. We also determine necessary and sufficient conditions on M and N for the existence of a simple graph G whose edge set can be partitioned into r cycles of lengths m 1 , m 2 , . . . , m r and also into t cycles of lengths n 1 , n 2 , . . . , n t . Such a graph is said to be (M; N )-decomposable.

“…Face 2‐colorability requires z ≡1, 3, 7, 9 (mod 12) in the nonorientable case, and z ≡3, 7 (mod 12) in the orientable case. In the articles , (see also for a survey), the authors constructed a ·2

^{italicbz^{2}}

nonisomorphic triangular embeddings of K z ( a and b being positive constants) in both the orientable and nonorientable cases, provided that z is sufficiently large and lies in certain congruence classes. For example, if z ≡7, 19 (mod 36), then it is shown in that, as z →∞, there are at least

face 2‐colorable triangular embeddings of K z in an orientable surface.…”

Section: Introductionmentioning

confidence: 99%

On the number of triangular embeddings of complete graphs and complete tripartite graphs

Grannell

Knor

2011

Self Cite

Abstract:We prove that for every prime number p and odd m>1, as s→∞, there are at least w w 2 (1/(p 4 m 2 )−o(1)) face 2-colorable triangular embeddings of K w,w,w , where w = m·p s . For both orientable and nonorientable embeddings, this result implies that for infinitely many infinite families of z, there is a constant c>0 for which there are at least z cz 2 nonisomorphic face 2-colorable triangular embeddings of K z .